Lectures on Etale Cohomology - J.S. Milne Top

Group Theory

Fields and Galois Theory

Algebraic Geometry

Algebraic Number Theory

Modular Functions and Modular Forms

Elliptic Curves -- see books.

Abelian Varieties

Lectures on Etale Cohomology

Class Field Theory

Complex Multiplication

Basic Theory of Affine Group Schemes

Lie Algebras, Algebraic Groups, and Lie Groups

Reductive Groups

Errata

pdf file for the current version (2.21)
### Contents

**The word étale.** There are two different words in French, "étaler", which means spread out or displayed and is used in
"éspace étalé", and "étale", which is rare except in poetry.
According to Illusie, it is the second that Grothendieck chose for étale morphism.
The Petit Larousse defines "mer étale" as "mer qui ne monte ni ne descend", i.e., the sea at the point of high or low
tide. For example, there is the quote from Hugo which I included in my book
"La mer était étale, mais le reflux commencait a se sentir".
I think Grothendieck chose the word because the way he pictured étale
morphisms reminded him of a calm sea at high tide under a full moon
(locally almost parallel bands of light, but not globally). I find
this image beautiful. A footnote in Mumford's Red Book on Algebraic
Geometry says: "The word apparently refers to the appearance of the
sea at high tide under a full moon in certain types of weather."

image### History

v2.01; August 9, 1998; first version on the web; 190 pages.

v2.10; May 20, 2008; corrected errors and improved the TeX; 196 pages. old version 2.10

v2.20; May 3, 2012; corrected; minor improvements; 202 pages.

v2.21; March 22, 2013; corrected; minor improvements; 202 pages.

These are the notes for a course taught at the University of Michigan in 1989 and 1998. In comparison with my book, the emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes. The notes also discuss the proof of the Weil conjectures (Grothendieck and Deligne).

- Introduction
- Etale Morphisms
- The Etale Fundamental Group
- The Local Ring for the Etale Topology
- Sites
- Sheaves for the Etale Topology
- The Category of Sheaves on
*X*_{et}. - Direct and Inverse Images of Sheaves.
- Cohomology: Definition and the Basic Properties
- Cech Cohomology
- Principal Homogeneous Spaces and
*H*^{1}. - Higher Direct Images; the Leray Spectral Sequence
- The Weil-Divisor Exact Sequence and the Cohomology of
*G*_{m} - The Cohomology of Curves
- Cohomological Dimension.
- Purity; the Gysin Sequence.
- The Proper Base Change Theorem.
- Cohomology Groups with Compact Support.
- Finiteness Theorems; Sheaves of
*Z*_{l}-modules - The Smooth Base Change Theorem.
- The Comparison Theorem.
- The Kunneth Formula.
- The Cycle Map; Chern Classes
- Poincare Duality
- Lefschetz Fixed-Point Formula.
- The Weil Conjectures.
- Proof of the Weil Conjectures, except for the Riemann Hypothesis
- Preliminary Reductions
- The Lefschetz Fixed Point Formula for Nonconstant Sheaves
- The MAIN Lemma
- The Geometry of Lefschetz Pencils
- The Cohomology of Lefschetz Pencils
- Completion of the Proof of the Weil Conjectures.
- The Geometry of Estimates

image

v2.10; May 20, 2008; corrected errors and improved the TeX; 196 pages. old version 2.10

v2.20; May 3, 2012; corrected; minor improvements; 202 pages.

v2.21; March 22, 2013; corrected; minor improvements; 202 pages.